A transmission problem for the Timoshenko system

Transcription

1 Volume 6, N., pp. 5 34, 7 Copyright 7 SBMAC ISSN A transmission problem for the Timoshenko system C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 3 Department of Mathematics, UFSJ, Praça Frei Orlano, São João el-rei, MG Department of Mathematics, UNESP, Rua Cristovão Colombo, São José o Rio Preto, SP 3 Department of Mathematics, UFPA, Rua o Una, Pará, PA s: raposo@ufsj.eu.br / walemar@ibilce.unesp.br / mlsantos@ufpa.br Abstract. In this work we stuy a transmission problem for the moel of beams evelope by S.P. Timoshenko ]. We consier the case of mixe material, that is, a part of the beam has friction an the other is purely elastic. We show that for this type of material, the issipation prouce by the frictional part is strong enough to prouce exponential ecay of the solution, no matter how small is its size. We use the metho of energy to prove exponential ecay for the solution. Mathematical subject classification: 35J55, 35J77, 93C. Key wors: transmission, Timoshenko, beams, exponential ecay, frictional amping. Introuction The transverse vibration of a beam is mathematically escribe by a system of two couple ifferential equations given by ρu tt (K (u x + ψ)) x =, I ρ ψ tt (E I ψ x ) x + K (u x + ψ) =, in (, L) (, ), in (, L) (, ). (.) Here, L is the length of the beam in its equilibrium position, t is the time variable an x is the space coorinate along the beam. The function u = u(x, t) #668/6. Receive: 3/I/7. Accepte: 9/II/7.

2 6 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM is the transverse isplacement of the beam an ψ = ψ(x, t) is the rotation angle of a filament of the beam. The coefficients ρ, I ρ, E, I an K are the mass per unit length, the polar moment of inertia of a cross section, Young s moulus of elasticity, the moment of inertia of a cross section an the shear moulus respectively. We enote ρ = ρ, ρ = I ρ, b = E I, k = K an we obtain irectly from (.) the following system ρ u tt k(u x + ψ) x =, ρ ψ tt b ψ xx + k(u x + ψ) =, in (, L) (, ), in (, L) (, ). (.) The mathematical moel escribing the vibrations of beam with fixe extremities is forme by the system (.), bounary conitions an initial ata u(, t) = u(l, t) = ψ(, t) = ψ(l, t) =, t >, u(, ) = φ, u t (, ) = φ, ψ(, ) = ψ, ψ t (, ) = ψ, in (, L). If friction is taken into account, the system (.) becomes ρ u tt k(u x + ψ) x + αu t =, ρ ψ tt b ψ xx + k(u x + ψ) + βψ t =, in (, L) (, ), in (, L) (, ), (.3) where α an β are positive constants (we assume α = β = ). The terms αu t an βψ t represent the attrition acting in the vertical vibrations an in the angle of rotation of the filaments of the beam, respectively. Dissipative properties associate to the system (.3) have been stuie by several authors by consiering issipative mechanism of frictional or viscoelastic type. The frictional issipation, obtaine by introuction of a frictional mechanism acting on the entire omain or on the bounary, was stuie in 7, 8, 9]. The viscoelastic issipation, given by a memory effect, was consiere in ] an 6]. An interesting problem comes out when the issipation acts only on a part of the omain. In the present paper we consier a frictional mechanism acting only on the part of the omain given by x L with < L < L. We prove that for every L the energy of the system ecays exponentially to zero as time Comp. Appl. Math., Vol. 6, N., 7

3 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 7 goes to infinity. In other wors, our result states that issipative properties of the system are transferre to the whole beam an stabilizes the system. The main result of this paper is Theorem an its corollary, both in section 5. The mathematical moel which eals with this situation is calle a transmission problem. From the mathematical point of view a transmission problem consist of an initial an bounary value problem for a hyperbolic equation for which the corresponing elliptic operator has iscontinuous coefficients. Hence, we can not expect to have regular solutions in the role omain. In the next section we establish the transmission problem an efine appropriately the notion of solution consiere. We use H m an L p to enote the usual Sobolev an Lebesgue spaces ]. The transmission problem In this section we escribe precisely the transmission problem treate in the paper an establish existence an regularity of solution. We begin by introucing the notation ρ j (x) = k(x) = b(x) = α = β = u(x, t) = ρ j, if x (, L ) ρ j, if x (L, L) k, if x (, L ) k, if x (L, L) b, if x (, L ) b, if x (L, L), if x (, L ), if x (L, L), if x (, L )], if x (L, L) u(x, t), if x (, L ) (, ) v(x, t), if x (L, L) (, ),,,,,, Comp. Appl. Math., Vol. 6, N., 7

4 8 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM ψ(x, t) = ψ(x, t), if x (, L ) (, ) φ(x, t), if x (L, L) (, ). Using the notation above, moel (.3) can be written in the following form: ρ u tt k (u x + ψ) x + u t =, in (, L ) (, ), (.) ρ ψ tt b ψ xx + k (u x + ψ) + ψ t =, in (, L ) (, ), (.) ρ v tt k (v x + φ) x =, in (L, L) (, ), (.3) ρ φ tt b φ xx + k (v x + φ) =, in (L, L) (, ), (.4) Dissipative part u(x), ψ(x) Elastic part v(x), φ(x) L L L with bounary conitions, u(, t) = v(l, t) = ψ(, t) = φ(l, t) =, t >, transmission conitions, k u(l, t) = k v(l, t), k u x (L, t) = k v x (L, t), ρ ψ t(l, t) = ρ φ t(l, t), an initial ata ρ u t(l, t) = ρ v t(l, t), k ψ(l, t) = k φ(l, t), b ψ x (L, t) = b φ x (L, t), (.5) u(, ) = u, u t (, ) = u, ψ(, ) = ψ, ψ t (, ) = ψ, in (, L ), v(, ) = v, v t (, ) = v, φ(, ) = φ, φ t (, ) = φ, in (L, L). (.6) We efine the notion of weak solution to the system (.)-(.6) as follows: Comp. Appl. Math., Vol. 6, N., 7

5 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 9 Definition. Let V, H m an L be the spaces efine by V = ( ˉw, w) H (, L ) H (L, L); ˉw() = w(l) =, ˉw(L ) = w(l ) H m = H m (, L ) H m (L, L) an L = L (, L ) L (L, L). We say that (u, v, ψ, φ) is a weak solution to the problem (.)-(.6) if for every ( ˉw, w) H (, T ; H V ) we have: ρ an + ρ ρ u t (x, T ) ˉw(x, T )x ρ L v t (x, T )w(x, T )x ρ u t (x, ) ˉw(x, )x u t (x, t) ˉw t (x, t)xt ρ L v t (x, )w(x, )x L v t (x, t)w t (x, t)xt + v t (x, t)w x (x, t)xt + k (u x + ψ)(x, t) ˉw x (x, t)xt L + k (v x + φ)(x, t)w x (x, t)xt = L ρ + ρ ρ + + ψ t (x, T ) ˉw(x, T )x ρ L φ t (x, T )w(x, T )x ρ + ψ t (x, ) ˉw(x, )x ψ t (x, t) ˉw t (x, t)xt ρ b ψ(x, t) x ˉw(x, t)xt + L φ t (x, )w(x, )x k (u x + ψ) ˉw(x, t)xt + b L k (v x + φ)(x, t)w(x, t)xt =. L φ t (x, t)w t (x, t)xt ψ t (x, t) ˉw x (x, t)xt L φ x (x, t)w x (x, t)xt The transmission problem for a single hyperbolic equation was stuie by Dautray an Lions 3], who prove the existence an regularity of solutions for Comp. Appl. Math., Vol. 6, N., 7

6 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM the linear problem. The existence an regularity of solutions to the transmission problem for the Timoshenko system is given in the following theorem: Theorem. If (u, v ), (ψ, φ ) V an (u, v ), (ψ, φ ) L, then there exists a unique weak solution (u, v, ψ, φ) to the system (.)-(.6) satisfying: (u, v), (ψ, φ) C(, ; V ) C (, ; L ). Moreover, if (u, v ), (ψ, φ ) H V an (u, v ), (ψ, φ ) V, then the weak solution is a strong solution an satisfies (u, v), (ψ, φ) C(, ; H V ) C (, ; V ) C (, ; L ). Proof. For the proof we procee in a quite similar manner as in 3]. The total energy associate to the system is efine by E(t) = + ρ u t + ρ ψ t + b ψ x + k u x + ψ x L ρ v t + ρ φ t + b φ x + k v x + φ x. Next we prove that the total energy associate to the system is ecreasing for every t >. Lemma. Let (u, v, ψ, φ) be the strong solution to the system (.)-(.6), then L t E(t) = u t x ψ t x. Proof. Multiplying (.) by u t an integrating by parts over the interval (, L ), we get ρ u t x = k (u x (L ) + ψ(l )) u t (L ) t (.7) k (u x + ψ) u tx x u t x. Comp. Appl. Math., Vol. 6, N., 7

7 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS Now, multiplying (.) by ψ t an integrating by parts over (, L ) we obtain ρ ψ t x + b ψ x x = b ψ x (L )ψ t (L ) t k (u x + ψ) ψ t x ψ t x. Multiplying (.3) by v t an integrating by parts on (L, L), we get ρ (.8) L v t x = k (v x + φ)(l )v t (L ) k (v x + φ) v tx x. (.9) t L L Multiplying (.4) by φ t an integrating by parts on (L, L) leas to ρ φ t x + b φ x x = b φ x (L )φ t (L ) t L L k (v x + φ) φ t x. L Now observe that an k k t φ x + ψ x = k (.) ( φ x + ψ ) ( φxt + ψ ) t x, (.) L φ t x + ψ ( x = k φ x + ψ ) ( φxt + ψ ) t x. (.) L L Summing up (.7), (.8), (.9) an (.), an using (.) an (.) together with the hypothesis of transmission we obtain L t E(t) = u t x ψ t x. L 3 Technical lemmas Now we evelop a series of technical results in orer to facilitate the proof of the main result of the paper. We begin by constructing a functional E(t), equivalent to the energy functional, which satisfies E(t) C E(), C <. In orer to o so, we use some multiplier techniques (usually associate to control problems) Comp. Appl. Math., Vol. 6, N., 7

8 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM an the following restrictions on the bounary conitions for the elastic part of the beam: v x (L) v x x. (3.) L L φ x (L) φ x x. (3.) L L Lemma. Let us efine E (t) = N E(t)+ x (ρ u xu t +ρ ψ xψ t ) x + x (ρ v xv t +ρ φ xφ t ) x. L Then t E (t) 4 k u x + b ψ x ] x 4 L k v x + b φ x ] x. Proof. Multiply (.) by x u x an integrate by parts over (, L ) to get L ρ t x u t u x x = L ρ u t(l ) ρ L u t x + L k u x(l ) k u x x + k x ψ x u x x x u t u x x. Multiply (.) by x ψ x an integrate by parts over (, L ) to obtain L ρ t x ψ t ψ x x = L ρ ψ t(l ) ρ L ψ t x + L b ψ x(l ) b + L k ψ(l ) k k ψ x x ψ x x ψ x u x x x ψ t ψ x x. (3.3) (3.4) Comp. Appl. Math., Vol. 6, N., 7

9 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 3 Multiplying (.3) by x v x an integrating by parts over (L, L) leas to L ρ t x v t v x x = L ρ L v t(l ) ρ v t x L + Lk v x(l) L k v x(l ) k L v x x + k L x φ x v x x. (3.5) Now multiply (.4) by x φ x an integrate over (, L ) to obtain L ρ t x φ t φ x x = L ρ L φ t(l ) ρ φ t x L + Lb φ x(l) L b φ x(l ) b k φ x x L k L φ(l ) L φ x x k L x φ x v x x. (3.6) Summing up (3.3), (3.4), (3.5), (3.6), an making use of the hypothesis on the transmission, the punctual terms are cancele an we get L x (ρ t u xu t + ρ ψ xψ t ) x + x (ρ t v xv t + ρ φ xφ t ) x L = Lk v x(l) + Lb φ x(l) k u x x xu t u x x b ψ x x + xψ t ψ x x k v x x b φ x x. L L Now using (3.), (3.) an the Young s inequality 4] we obtain L x (ρ t u xu t + ρ ψ xψ t ) x + x (ρ t v xv t + ρ φ xφ t ) x L k 4 L v x x k L v x x + b 4 L φ x x b L φ x x Comp. Appl. Math., Vol. 6, N., 7

10 4 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM k + ˉC Now, if we efine E (t) = N E(t)+ u x x + k 4 ψ t x + u t x an choose N > ˉC we conclue t E (t) 4 u x x b ]. ψ x x + b 4 ψ x x x ( ρ u xu t +ρ ψ ) L xψ t x + x ( ρ v xv t +ρ φ ) xφ t x k u x + b ψ x ] x 4 L L k v x + b φ x ] x. It is worth noticing that the estimate above is important in two aspects. First, it recovers a part of energy with minus sign. Secon, it will play a funamental role in the next two lemmas controlling punctual terms which will come up in the search for other negative terms of the energy. Lemma 3. Define Then Proof. E (t) = N E(t) + ρ x u t(u x + ψ) x. t E (t) ρ L u t(l ) + k L u x(l ) + ψ(l ) C ut + ψ t + u x + ψ ] x. Multiply (.) by x (u x +ψ) an integrate by parts over (, L ) to obtain L x ρ t (u x + ψ)u t = ρ L u t(l ) ρ L u t x + k L u x(l ) + ψ(l ) k L + ρ x u t ψ t x x u t (u x + ψ) x. u x + ψ x Comp. Appl. Math., Vol. 6, N., 7

11 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 5 Using Young s inequality, we get t k L 4 Now, efining we obtain k L 4 x ρ (u x + ψ)u t ρ L u t(l ) + k L u x(l ) + ψ(l ) u x + ψ x + C u t x + C E (t) = N E(t) + ρ x u t(u x + ψ) x t E (t) ρ L u t(l ) + k L u x(l ) + ψ(l ) u x + ψ x + ( C N ) u t x + ( C N ) ψ t x. If we choose N > C we conclue that there exists C > such that an then Lemma 4. k L 4 E 3 (t) = ρ L ρ Then t E (t) ρ L u t(l ) + k L u x(l ) + ψ(l ) u x + ψ x C u t x C ψ t x, t E (t) ρ L u t(l ) + k L u x(l ) + ψ(l ) C Let E 3 be efine as L x (ρ v xv t +ρ φ xφ t ) x + ut + ψ t + u x + ψ ] x. ψ t x. L (x L) ρ v t(v x + φ) ] x. t E 3(t) ρ (L L ) v t (L ) + k (L L ) v x (L ) + φ(l ) C vt + φ t + v x + φ ] x. Comp. Appl. Math., Vol. 6, N., 7

12 6 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM Proof. Multiply (.3) by (x L)(v x + φ) an integrate by parts over (L, L) to get L ] (x L) ρ t v t(v x + φ) L Using Young s inequality we obtain x = ρ (L L ) v t (L ) ρ L v t x + k (L L ) v x (L ) + φ(l ) k + ρ L v x + φ x L (x L)v t φ t x. L (x L) ρ t v t(v x + φ) ] x ρ (L L ) v t (L ) L + k (L L ) v x (L ) + φ(l ) ρ v t x + ρ v t x Now we set k v x + φ x + Lρ L L φ t x. E 3 (t) = ρ L ρ an verify that L x (ρ v xv t + ρ φ xφ t ) x + L (x L) ρ v t(v x + φ) ] x t E 3(t) ρ (L L ) v t (L ) + k (L L ) v x (L ) + φ(l ) k v x + φ x L ρ ρ L ρ ( L ρ ) + L ρ φ t x. L L v t x Comp. Appl. Math., Vol. 6, N., 7

13 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 7 It follows then t E 3(t) ρ (L L ) v t (L ) + k (L L ) v x (L ) + φ(l ) C v t + φ t + v x + φ ] x. Observe that in the attempt to recover the total energy of the system with negative sign we introuce the lemmas 3 an 4. Now we nee to control them. It will be achieve with the ai of the next section. 4 Compactness This section is eicate to iscuss the argument of compactness employe in the proof of the main result of the paper. First we introuce a notation; the symbol is use to enote convergence in the norm of the Sobolev space L as in 5]. For sake of completeness, we state the following result ue to J.U. Kim. Lemma 5. Let (u k ) be a sequence of functions satisfying u k u u k t u t in L (, T, H β (, L)), in L (, T, H α (, L)), as k, with α < β. Then u k u in C(, T ], H r (, L)), for some r < β. Proof. See 5]. Lemma 6 (Lemma of compactness). If we efine B(L, t) = ρ L u t(l ) + k L u x(l ) + ψ(l ) + ρ (L L ) v t (L ) + k (L L ) v x (L ) + φ(l ) Comp. Appl. Math., Vol. 6, N., 7

14 8 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM then, for every η > there exists a constant C η > inepenent of the initial ata, such that B(L, t)t η E(t)t + C η ux + ψ x ] xt + vx + φ x ] xt L for every strong solution (u, v, ψ, φ) to the system (.)-(.6) an sufficiently large T. Proof. We use a contraiction argument. Define B n (L, t) = ρ L un t (L ) + k L un x (L ) + ψ n (L ) + ρ (L L ) Suppose that there exists a sequence of initial ata vt n (L ) + k (L L ) vx n (L ) + φ n (L ). (u n, ψ n ) H V, (v n, φn ) H V, (u n, ψ n ) V, (vn, φn ) V, an a positive constant η > such that the corresponing solution (u n, ψ n ), (v n, φ n ) of the problem ρ un tt k (u n x + ψ n ) x + u n t = in (, L ) (, ), ρ ψ n tt b ψ n xx + k (u n x + ψ n ) + ψ n t = in (, L ) (, ), ρ vn tt k (v n x + φn ) x = ρ φn tt b φ n xx + k (v n x + φn ) = with bounary conitions, in (L, L) (, ), in (L, L) (, ), u n (, t) = v n (L, t) = ψ n (, t) = φ n (L, t) =, t >, transmission conitions, k u n (L, t) = k v n (L, t), k u n x (L, t) = k v n x (L, t), ρ ψ n t (L, t) = ρ φn t (L, t), ρ un t (L, t) = ρ vn t (L, t), k ψ n (L, t) = k φ n (L, t), b ψ n x (L, t) = b φ n x (L, t), Comp. Appl. Math., Vol. 6, N., 7

15 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 9 an initial ata u n (, ) = u n, u t(, ) = u n, ψn (, ) = ψ n, ψn t (, ) = ψn in (, L ), v n (, ) = v n, vn t (, ) = vn, φn (, ) = φ n, φn t (, ) = φn in (L, L), satisfies an the following inequality > η E n (t)t + n + Then the integral an also, an B n (L, t)t =, n N, (4.) L v n x + φ n x ] x t u n x + ψ n x ] x t. E n (t)t is boune for every n N, u n x + ψ n x ] x t as n, (4.) L v n x + φ n x ] x t as n. (4.3) Now we observe that E n (t) > an that E n (t)t is boune. Hence E n (t) is boune an we can take a subsequence of (u n, ψ n ), (v n, φ n ) (for which we use the same notations) such that u n u in L (, T, H V ), ψ n ψ in L (, T, H V ), v n v in L (, T, V ), φ n φ in L (, T, V ). Applying the lemma 5 we conclue that for r < u n u in C(, T ]; H r (, L )), ψ n ψ in C(, T ]; H r (, L )), v n v in C(, T ]; H r (L, L)), φ n φ in C(, T ]; H r (L, L)). Comp. Appl. Math., Vol. 6, N., 7

16 3 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM It follows from (4.) that B(L, t) =. (4.4) We observe that the convergences (4.) an (4.3) result in u x = almost everywhere in (, L ) (, T ), ψ x = almost everywhere in (, L ) (, T ), v x = almost everywhere in (L, L) (, T ), φ x = almost every where in (L, L) (, T ). Now, applying Poincare s inequality we obtain This estimates implies u(l ) t c p u x (L ) t c p ψ(l ) t c p v(l ) t c p v x (L ) t c p φ(l ) t c p B(L, t) =, u x xt =, u xx xt =, ψ x xt =, L v x xt =, L v xx xt =, L φ x xt =. which is a contraiction to (4.4). This completes the proof of the lemma. We are now reay to prove the main result of this paper, that is, the exponential ecay of the energy associate to the transmission problem for the Timoshenko System with frictional issipation. This is the content of the next section. 5 Exponential ecay Theorem. Let (u, ψ, v, φ) be a strong solution to the transmission problem for the Timoshenko System efine by (.)-(.6). Then there exist positive Comp. Appl. Math., Vol. 6, N., 7

17 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 3 constants C an w such that E(t) C E()e ωt. Proof. We start efining E(t) = N 3 E (t) + E (t) + E 3 (t). It follows from lemmas, 3, an 4 L t E(t) C E(t) C N 3 ux + ψ x ] x + vx + φ x ] x. + B(L, t). L Now, integrating this inequality over (, T ) an using the Lemma of Compactness we obtain E(T ) E() C E(t) t + η E(t) t C N C η + ux + ψ x ] x t vx + φ x ] x t L ux + ψ x ] xt L vx + φ x ] xt If we choose an fix η < C an N 3 such that N 3 C > C η, we get. Since E(t) ecreases, we have E(T ) E() C E(t)t. (5.) Comp. Appl. Math., Vol. 6, N., 7 T E(T ) E(t)t. (5.)

18 3 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM Using (5.) in (5.) we obtain E(T ) E() T C E(T ). Now observe that for sufficiently large N we have from what follows that or else N E(t) E(t) N E(t), (5.3) E(T ) E() C N T E(T ), E(T ) α E() with α = + C ]. N Note that α oes not epen on the initial ata, an hence, by using the semigroup property we have E(t + T ) α E(t) for every t >. (5.4) For t >, there exists a natural n an a real r, r < T such that t = nt + r. This is equivalent to n = t T r T. Now, using the inequalities (5.3) an (5.4) n times we obtain E(t) α n E(r) Nα n E(r). Observing once more that E(t) ecreases we have ( ) where w = ln α T. Finally, using (5.3) we obtain E(t) Nα ( t T r T ) E() Nα E() e w t, E(t) 4α E() e w t, an conclue the proof. We can exten the previous theorem to the weak solutions by using simple ensity argument an the laws of semi-continuity for the energy functional. In this irection we have the following corollary. Comp. Appl. Math., Vol. 6, N., 7

19 C.A. RAPOSO, W.D. BASTOS an M.L. SANTOS 33 Corollary. Uner the hypothesis of the previous theorem, there exist positive constants C an w, such that E(t) C E()e wt, for every weak solution (u, ψ, v, φ) of the system (.)-(.6). 6 Concluing remarks During the past several ecaes, many authors have stuie the same physical phenomenon for the Timoshenko system formulate into ifferent mathematical moels. Our approach to this problem is important not only from mathematical but mainly from the physical point of view with applications in Mechanics, amongst other branches of science. The system stuie here is a moel for vibrating beams subjecte to two frictional mechanisms. More precisely, we prove that the presence of two frictional amping acting in a natural way on a small part of the beam, is enough to stabilize the whole beam. Moreover, it stabilizes quickly (at exponential rate). To the best of our knowlege, our result is the first in this irection. In this sense, this work generalizes the results previously obtaine for Timoshenko s system where attrition acting in the whole beam was consiere. REFERENCES ] R.A. Aams, Sobolev Spaces, Acaemic Press, New York, (975). ] F. Ammar-Khoja, A. Benaballah, J.E.M. Rivera an R. Racke, Energy ecay for Timoshenko systems of memory type. J. Differential Equations, 94() (3), ] R. Dautray an J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol., Masson S.A., Paris (984). 4] G.H. Hary, J.E. Littlewoo an G. Polya, A Theorem of W. H. Young 8.3 in Inequalities, n e., Cambrige University Press, Cambrige, (988), pp ] J.U. Kim, A Bounary Thin Obstacle Problem for a Wave Equation. Comm. Partial Differential Equations, 4 (8 & 9) (989), 6. 6] Z. Liu an C. Peng, Exponential stability of a viscoelastic Timoshenko Beam. Av. Math. Sci. Appl., 8 (998), ] C.A. Raposo, J. Ferreira, M.L. Santos an N.N.O. Castro, Exponential Stability for the Timoshenko System With Two Weak Damping. Appl. Math. Lett., 8 (5), Comp. Appl. Math., Vol. 6, N., 7

20 34 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM 8] Dong-Hua Shi an De-Xing Feng, Exponential ecay of Timoshenko beam with locally istribute feeback. IMA J. Math. Control Inform., 8 (), ] Dong-Hua Shi, De-Xing Feng an Qing-Xu Yan, Feeback stabilization of rotating Timoshenko beam with aaptive gain. Int. J. Control, 74(3) (), ] S.P. Timoshenko an J.M. Gere, Mechanics of Materials, D. Van Nostran Company, Inc, New York (97). Comp. Appl. Math., Vol. 6, N., 7